With sufficiently mild Brownian kicking the working efficiency of the active particle is reduced by viscous drag, but the basic operation of the active particle persists. Stronger thermal fluctuations from gas collisions can indeed disrupt the Hamiltonian mechanism, however. To show this we present two different numerical evolutions in Fig. REF , with the gas particle mass \(m\)  differing by an order of magnitude; the horizontal axis is again the dimensionless \(-\lambda /I_0\)  rescaled time of the dissipationless case, which is also shown again for comparison. The vertical axis here is neither \(I\)  nor \(p\)  but rather the action variable \(S\) , defined as the area enclosed in the \(\varphi ,I\)  phase-space plane by the contour on which the instantaneous function \(H^{\prime }\)  equals the instantaneous value of \(H^{\prime }\) . The quantity \(A_0=4\pi I_0\)  is the total area of the phase space. The fact that in Fig. REF  the blue curve of the dissipationless case oscillates around a constant value represents the well-known adiabatic invariance of the action [1]}. All lines correspond to the same initial values \(I_i/I_0=0.85\) , \(p_i/p_\text{c}=1\) , and \(\alpha _i=\frac{5}{8}\pi \) . The collision properties for the red line are \(\sigma =0.15 \frac{p_c}{M}\) , \(\mu =\frac{2}{101}\) , \(\bar{\tau }=\mu \frac{k I_0}{f}\) , and \(\sigma =0.1 \frac{p_c}{M}\) , \(\mu =\frac{2}{1001}\) , \(\bar{\tau }=\mu \frac{k I_0}{f}\)  for the orange line.
<FIGURE>